Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept,and when I want to learn something new, I make it my first job to build one.

The beginner should not be discouraged if he finds he does not have the prerequisites for reading the prerequisites.

When a student comes and asks, "Should I become a mathematician?" the answer should be no. If you have to ask, you shouldn't even ask.

If the NSF had never existed, if the government had never funded American mathematics, we would have half as many mathematicians as we now have, and I don't see anything wrong with that.

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

The best way to learn is to do; the worst way to teach is to talk.